In a radio communication system, problems of frequency offset and sampling time error between the transmitter and the receiver usually occur, which degrade the performance of the system. Not only the frequency discrepancy between the oscillators of the transmitter and the receiver but also the Doppler effect will result in frequency offset. To non-coherently demodulate the modulated signal sent from the transmitter and avoid degrading the performance of the communication system, one still needs to obtain the correct frequency offset and sampling time error from the signals and respectively let the frequency offset and the sampling time error be compensated and modified. And when the receiver samples the modulated signals, a preferred sampling time is prerequisite for obtaining correct demodulated signals.
Further, due to the fact that the radio data are often transmitted in a burst mode, one may try to find a fast and stable method to estimate the mentioned frequency offset and sampling time error and then modify or compensate the frequency offset of the system in order to maintain the system performance.
A paper of Ralf Mehlan et al. titled "A Fully Digital Feedforward MSK Demodulator with Joint Frequency Offset and Symbol Timing Estimation for Burst Mode Mobile Radio", IEEE Transactions on Vehicular Technology, Vol. VT-42, No. 4, November 1993, pp. 434-443 discloses that a demodulating system for demodulating the signals sent from an MSK transmitter is to delay the received signal, multiply the delayed signal with the received signal, square the mentioned multiplication, and then obtain the sampling time error and the frequency offset by using several sets of filters. To achieve an optimal receiving performance, the system requires at least four sets of filters of length 64, which makes the implementation of the demodulating system uneconomical.
The Gaussian Low-Pass Filtered Minimum Shift Keying transmitter (also named GMSK) is now widely applied by many personal and mobile communicating system standards, which includes a Gaussian Low-Pass filter and a Frequency Modulator (FM modulator). If the GMSK transmitter is without the Gaussian Low-Pass filter, then it becomes a Minimum Shift Keying transmitter (MSK transmitter). But, a structure of the GMSK demodulating system using Fast Fourier Transform operation for estimating frequency offset and sampling error has heretofore not been used.